On the Decidability of Connectedness Constraints in 2D and 3D Euclidean Spaces

We investigate (quantifier-free) spatial constraint languages with equality, contact and connectedness predicates as well as Boolean operations on regions, interpreted over low-dimensional Euclidean spaces. We show that the complexity of reasoning varies dramatically depending on the dimension of the space and on the type of regions considered. For example, the logic with the interior-connectedness predicate (and without contact) is undecidable over polygons or regular closed sets in the Euclidean plane, NP-complete over regular closed sets in three-dimensional Euclidean space, and ExpTime-complete over polyhedra in three-dimensional Euclidean space.Comment: Accepted for publication in the IJCAI 2011 proceeding

Modal Logics of Topological Relations

Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the Egenhofer-Franzosa (or RCC8) relations between regions in topological spaces such as the real plane. We investigate the expressive power and computational complexity of logics obtained in this way. It turns out that our modal logics have the same expressive power as the two-variable fragment of first-order logic, but are exponentially less succinct. The complexity ranges from (undecidable and) recursively enumerable to highly undecidable, where the recursively enumerable logics are obtained by considering substructures of structures induced by topological spaces. As our undecidability results also capture logics based on the real line, they improve upon undecidability results for interval temporal logics by Halpern and Shoham. We also analyze modal logics based on the five RCC5 relations, with similar results regarding the expressive power, but weaker results regarding the complexity

A Modal Logic for Subject-Oriented Spatial Reasoning

We present a modal logic for representing and reasoning about space seen from the subject\u27s perspective. The language of our logic comprises modal operators for the relations "in front", "behind", "to the left", and "to the right" of the subject, which introduce the intrinsic frame of reference; and operators for "behind an object", "between the subject and an object", "to the left of an object", and "to the right of an object", employing the relative frame of reference. The language allows us to express nominals, hybrid operators, and a restricted form of distance operators which, as we demonstrate by example, makes the logic interesting for potential applications. We prove that the satisfiability problem in the logic is decidable and in particular PSpace-complete

Rendiconti dell'Istituto di Matematica dell'Università di Trieste. An International Journal of Mathematics. Vol. 44 (2012)

Rendiconti dell’Istituto di Matematica dell’Università di Trieste was founded in 1969 by Arno Predonzan, with the aim of publishing original research articles in all fields of mathematics and has been the first Italian mathematical journal to be published also on-line. The access to the electronic version of the journal is free. All published articles are available on-line. The journal can be obtained by subscription, or by reciprocity with other similar journals. Currently more than 100 exchange agreements with mathematics departments and institutes around the world have been entered in

Undecidability of Plane Polygonal Mereotopology

This paper presents a mereotopological model of polygonal regions of the Euclidean plane and an undecidability proof of its firstorder theory. Restricted to the primitive operations the model is a Boolean algebra. Its single primitive predicate defines simple polygons as the topologically simplest polygonal regions. It turns out that both the relations usually provided by mereotopologies and more subtle topological relations are elementarily definable in the model. Using these relations, Post's correspondence problem, known as undecidable, can be reduced to the decision problem of the model. 1 Introduction Formalizing commonsense concepts of space has received much attention both in the philosophical literature and in recent AI research. Mereotopological theories as well as most calculi for spatial reasoning deal with spatial regions, i.e. the parts of space occupied by physical bodies, and their topological relations as intuitive concepts of our commonsense space. Whereas mereotopolo..